Exploratory Analysis 1

Introduction to spatial R packages and plotting functions

Introduction

Crucial to all good statistical analyses is a thorough exploratory analysis. In this document, we will introduce a few techniques for developing an understanding of our dataset, including an understanding of its limitations.

Load packages

Let’s load all the required R packages. See our tutorial 0_Installing_packages.html for installation information.

library(rgeos)
library(rgdal)
library(sp)
library(maptools)
library(spatstat)
library(spdep)
library(INLA)
library(inlabru)
library(readxl)
library(lubridate)
library(ggmap)
library(raster)

File set up

If you have not done so already, you need to download the two set of files associated with the workshop.

First, all the workshop script and tutorial files can be found on Github at github.com/joenomiddlename/DFO_SDM_Workshop_2020. To download these files on your computer, press on Code and then Download ZIP. Once downloaded, unzip the DFO_SDM_Workshop_2020-main.zip file. To be able to access some of the precompiled data, we need to make the unzipped folder the working directory. To do so in R Studio, navigate to the correct folder using the bottom right panel (folder view, you can use … button) and open it. Then, click “Set as Working Directory” under the tab ‘More’.

We should be inside the folder titled: ‘DFO_SDM_Workshop_2020’, you can verify this by using getwd().

Now we can load in the precompiled data

list2env(readRDS('./Data/Compiled_Data.rds'), globalenv())

## <environment: R_GlobalEnv>

Second, to help play with model results quickly, we need to download the pre-compiled modelling result files, which can be found here.

Fin whale data

The data we use throughout this workshop is data on fin whales. We use two main data sources:

• sightings from systematic DFO?? boat surveys

• sightings from two? whale-watching operators

The sightings are from 201?. The sightings from the systematic survey, but not those of the whale-watching operators, are accompanied with the boat trackline.

Coordinate reference system

These data are spatial data. The first thing we must always do is check for consistency between the coordinate reference systems (CRS) of each spatial object in use! To print the CRS of an sp object, simply add @proj4string to the name of the spatial object and run. For example, to check the CRS of the WW sightings we run:

Sightings_DRWW_sp@proj4string 

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

We check the CRS of the other objects, but omit the code for brevity.

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

## CRS arguments: +proj=longlat +datum=WGS84 +no_defs

All of the spatial objects are in lat/lon - good! For future analysis we will be projecting the data into a different coordinate reference system to better preserve euclidean distance.

Finally, let’s turn off all warnings associated with coordinate reference systems.

rgdal::set_rgdal_show_exportToProj4_warnings(FALSE)
rgdal::set_thin_PROJ6_warnings(TRUE)
options("rgdal_show_exportToProj4_warnings"="none")

Plotting the data

Our first goal is generally to plot the data on a map. The gg() and gmap() functions from the inlabru package will prove extremely useful at plotting spatial data! Our spatial objects we will from the sp package. The class of objects from the sp package begin with ‘Spatial’. For example SpatialPointsDataFrame.

We have written a bespoke function gg.spatiallines_mod() to easily add SpatialLinesDatFrame objects to the plots too. This will prove useful for plotting transect lines. We load the bespoke functions to the working environment now.

source('utility_functions.R')

Let’s plot our data!

If the data are in lat/lon format then the gmap() function will automatically add a terrain layer from Google Maps to the plots. Let’s plot the survey sightings in blue, the survey tracklines in black, the whale watch sightings in purple, the whale watch ports in red.

gmap(Sightings_survey) +
  gg(Domain) +
  gg.spatiallines_mod(Effort_survey) +
  gg(Sightings_survey, colour='blue') +
  gg(Sightings_DRWW_sp, colour='purple') +
  gg(WW_ports, colour='red')

## Warning in proj4string(x): CRS object has comment, which is lost in output

By default gmap uses Google maps, see Additional tips for ways to use open-source maps for publication.

To remove the map layer, simply replace the gmap(Sightings_survey) with ggplot().

ggplot() + # Notice the empty ggplot() call
  gg(Domain) +
  gg.spatiallines_mod(Effort_survey) +
  gg(Sightings_survey, colour='blue') +
  gg(Sightings_DRWW_sp, colour='purple') +
  gg(WW_ports, colour='red')

This plot hides some crucial information regarding the data collection. For example, the survey sightings and tracklines do not come from a single survey, or even a single organisation! Let’s plot this! The easiest way to do this is to subset the data accordingly!

table(Effort_survey$DATASET)

## 
##    DFO NOAA_1 NOAA_2 
##     49     54     70

# there are 3 surveys
ggplot() +
  gg(Domain) +
  gg.spatiallines_mod(Effort_survey[Effort_survey$DATASET=='DFO',], colour='purple') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$DATASET=='NOAA_1',], colour='red') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$DATASET=='NOAA_2',], colour='yellow')

This is problematic! The DFO tracklines (in purple) do not overlap with the two NOAA surveys! Thus, any future model will be unable to identify any differences in protocol efficiency. This is because any model intercepts will be confounded with the latent spatial field, but more on that later!

Exercise 1

In addition, the surveys were conducted across 4 separate years. Let’s plot the survey tracklines by year. Note that the names of the variables in the Effort_Survey are: DATASET and YEAR. Try this on your own! If you get stuck, click ‘Show Answer Code’. Hint: The YEAR variable is of type character and contains 4 unique values (see below).

table(Effort_survey$YEAR)

## 
## 2007 2008 2009 2011 
##  101   20   19   33

class(Effort_survey$YEAR)

## [1] "character"

ggplot() +
  gg(Domain) +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2007',], colour='purple') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2008',], colour='red') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2009',], colour='blue') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2011',], colour='yellow')

## Regions defined for each Polygons

Note that the surveys from years 2007, 2008 and 2009 covered largely different regions! Again, this is problematic if we want to model any changes in the whale distribution over time! The effect of year will be confounded by the spatial field. That being said, the data from 2011 appear to be a good candidate for model comparison as the spatial range overlaps with the other 3 years’ effort. We will holdout this data as our test data and use 2007, 2008, and 2009 as our training data.

xtabs(~ YEAR + DATASET, data=Effort_survey@data)

##       DATASET
## YEAR   DFO NOAA_1 NOAA_2
##   2007  49      3     49
##   2008   0     20      0
##   2009   0     19      0
##   2011   0     12     21

2011’s data comes exclusively from NOAA.

Transforming the data into a new CRS

For modelling, we will transform the data from lat/lon into a new “Canadian” CRS: NAD83 datum with UTM Zone 20N projection. This projection will help to preserve euclidean distance between points. The EPSG code is 2961.

To do the transformation, we will use the spTransform() function. For example, to transform the Whale Watch sightings spatial object Sightings_DRWW_sp, we first define the CRS object as follows:

Can_proj <- CRS("+init=EPSG:2961")
Can_proj <- fm_crs_set_lengthunit(Can_proj, unit='km')

The second line of code specifies that we want to work in units of km instead of the default meters. This can prove vital in applications to avoid numerical overflow.

Next, we transform Sightings_DRWW_sp:

Sightings_DRWW_sp <- spTransform(Sightings_DRWW_sp, Can_proj)

## Warning in proj4string(x): CRS object has comment, which is lost in output

Sightings_DRWW_sp@proj4string

## CRS arguments:
##  +init=EPSG:2961 +proj=utm +zone=20 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0
## +units=km +no_defs

Notice the changed output from calling @proj4string. We repeat this for all the spatial objects that are points or lines.

Sightings_Opp_sp <- spTransform(Sightings_Opp_sp, Can_proj)

## Warning in proj4string(x): CRS object has comment, which is lost in output

Sightings_survey <- spTransform(Sightings_survey, Can_proj)

## Warning in proj4string(x): CRS object has comment, which is lost in output

Effort_survey <- spTransform(Effort_survey, Can_proj)

## Warning in proj4string(x): CRS object has comment, which is lost in output

WW_ports <- spTransform(WW_ports, Can_proj)

## Warning in proj4string(x): CRS object has comment, which is lost in output

Transforming the ‘raster’-like SpatialPixelsDataFrame objects (Slope, Bathym, Dist_Brier, and Dist_Quoddy) using spTransform would be inappropriate here. The projection leads to a curvature of the pixels. A more appropriate approach here is to use bilinear interpolation. The projectRaster() function from the raster package works great for this. This requires converting the SpatialPixelsDataFrame object into an object of type raster. This is made easy with the function raster(). Finally, to convert the raster object back into a SpatialPixelsDataFrame, we can use the as() function from the maptools package. This function is extremely useful for converting spatial objects between the popular packages: sp, spatstat, and sf. We use this function substantially throughout the workshop.

Slope <- raster(Slope)
Slope <- projectRaster(Slope, crs=Can_proj)
Slope <- as(Slope, 'SpatialPixelsDataFrame') # Note the specification of class
# repeat for Bathym, combining into one single function call
Bathym <- as(projectRaster(raster(Bathym), crs=Can_proj), 'SpatialPixelsDataFrame')
Domain <- spTransform(Domain, Can_proj)

## Warning in proj4string(x): CRS object has comment, which is lost in output

Dist_Brier <- as(projectRaster(raster(Dist_Brier), crs=Can_proj), 'SpatialPixelsDataFrame')
Dist_Quoddy <- as(projectRaster(raster(Dist_Quoddy), crs=Can_proj), 'SpatialPixelsDataFrame')

Plot the (transformed) Bathymetry, Slope, and Distance from Port spatial objects. We are going to combine these into a single plot using the multiplot() function from the inlabru package. This function takes as input ggplot objects and an argument layout, specifying how the plots should be arranged (see Additional tips for ways to change the layout).

multiplot(ggplot() +
  gg(Domain) +
  gg(Bathym) + xlab('East(km)') + ylab('North(km)') + labs(fill='Bathymetry') + 
    coord_fixed(ratio = 1),
ggplot() +
  gg(Domain) +
  gg(Slope) + xlab('East(km)') + ylab('North(km)') + labs(fill='Slope') + 
  coord_fixed(ratio = 1),
ggplot() +
  gg(Domain) +
  gg(Dist_Brier) + xlab('East(km)') + ylab('North(km)') + 
  coord_fixed(ratio = 1),
ggplot() +
  gg(Domain) +
  gg(Dist_Quoddy) + xlab('East(km)') + ylab('North(km)')+ 
  coord_fixed(ratio = 1),
layout=matrix(1:4, nrow=2, ncol=2, byrow = TRUE))

Don’t like the colour scheme? We can define our own manually!

colsc <- function(...) {
  scale_fill_gradientn(colours = rev(RColorBrewer::brewer.pal(11,"PuBuGn")),
                       limits = range(...))
}

Look at ?RColorBrewer::brewer.pal to see what other colour palettes are available.

multiplot(ggplot() +
  gg(Domain) +
  gg(Bathym) + xlab('East(km)') + ylab('North(km)') + labs(fill='Bathymetry') +
  colsc(Bathym@data[,1]),
ggplot() +
  gg(Domain) +
  gg(Slope) + xlab('East(km)') + ylab('North(km)') + labs(fill='Slope') +
  colsc(Slope@data[,1]),
ggplot() +
  gg(Domain) +
  gg(Dist_Brier) + xlab('East(km)') + ylab('North(km)') +
  colsc(Dist_Brier@data[,1]),
ggplot() +
  gg(Domain) +
  gg(Dist_Quoddy) + xlab('East(km)') + ylab('North(km)') +
  colsc(Dist_Quoddy@data[,1]),
layout=matrix(1:4, nrow=2, ncol=2, byrow = T))

## Warning in RColorBrewer::brewer.pal(11, "PuBuGn"): n too large, allowed maximum for palette PuBuGn is 9
## Returning the palette you asked for with that many colors

## Warning in RColorBrewer::brewer.pal(11, "PuBuGn"): n too large, allowed maximum for palette PuBuGn is 9
## Returning the palette you asked for with that many colors

## Warning in RColorBrewer::brewer.pal(11, "PuBuGn"): n too large, allowed maximum for palette PuBuGn is 9
## Returning the palette you asked for with that many colors

## Warning in RColorBrewer::brewer.pal(11, "PuBuGn"): n too large, allowed maximum for palette PuBuGn is 9
## Returning the palette you asked for with that many colors

Exercise 2

Have a go at creating your own colour palette function. Investigate the effects of changing both arguments to brewer.pal.

colsc2 <- function(...){
  scale_fill_gradientn(colours = rev(RColorBrewer::brewer.pal(7,"Spectral")),
                       limits = range(...))
}
multiplot(ggplot() +
  gg(Domain) +
  gg(Bathym) + xlab('East(km)') + ylab('North(km)') + labs(fill='Bathymetry') +
  colsc2(Bathym@data[,1]),
ggplot() +
  gg(Domain) +
  gg(Slope) + xlab('East(km)') + ylab('North(km)') + labs(fill='Slope') +
  colsc2(Slope@data[,1]),
ggplot() +
  gg(Domain) +
  gg(Dist_Brier) + xlab('East(km)') + ylab('North(km)') +
  colsc2(Dist_Brier@data[,1]),
ggplot() +
  gg(Domain) +
  gg(Dist_Quoddy) + xlab('East(km)') + ylab('North(km)') +
  colsc2(Dist_Quoddy@data[,1]),
layout=matrix(1:4, nrow=2, ncol=2, byrow = T))

Testing model assumptions and investigating trends

JOE: we need to define all of the terms used later in the tutorial, e.g., \(\lambda_{eff}\). Depending on the lecture, you can be or less technical. But they should all be described at least intuitively here up-front

We will be fitting log-Gaussian Cox process models to the sightings data. This model assumes that, conditional upon the density surface \(\lambda_{obs}(\textbf{s})\), the sighting locations are independent.

Planned surveys can be designed to (approximately) satisfy the independence assumption. The boat speed can be fixed at a high value relative to the individuals’ speeds and a narrow perpendicular field-of-view can be enforced. Then, assuming that the individuals of the target species are in equilibrium with respect to their stationary distribution \(\lambda_{true}(\textbf{s})\) (i.e. the species’ density) and locally mixing faster than the time between revisits by observers, the independence assumption may reasonably hold.

Conversely, no such assumptions can be made regarding the whale-watch sightings. The unknown positions of the whale-watching boats, the 360 degree fields-of-view on-board, the variable boat speed, and the repeated sightings of the same individuals throughout each day can all invalidate the independence assumption required to fit a log-Gaussian Cox process.

In our workshop, we will see some potential solutions to address this issue of independence. In this session, we will perform some exploratory analyses to investigate the suitability of the independence assumption, and to investigate possible ways of estimating observer effort.

Investigating the autocorrelation between whale-watch sighting locations

Let’s look at the whale-watching data. Specifically, let’s investigate the 2nd, 3rd and 4th sightings in the dataset. Notice that they were made on the same day and very close together in time. Often, little information is provided on the database management procedures. For example, are these multiple encounters with the same individual? Would the database discard or retain repeated sightings?

Sightings_DRWW_sp@data[2:4,c('WS_TIME','WS_DATE','PLATFORM')]

## # A tibble: 3 x 3
##   WS_TIME    WS_DATE             PLATFORM               
##   <Period>   <dttm>              <chr>                  
## 1 16H 5M 0S  2007-06-19 00:00:00 BRIER ISLAND WHALEWATCH
## 2 16H 23M 0S 2007-06-19 00:00:00 BRIER ISLAND WHALEWATCH
## 3 16H 32M 0S 2007-06-19 00:00:00 BRIER ISLAND WHALEWATCH

Without additional information available on the database management protocols, we can investigate whether these sightings could be of the same individual. To investigate this hypothesis, we can compute the distance between the sightings using the gDistance() function, the time between the sightings, and the required travel speed of the individual under the assumed hypothesis.

# How far away are these sightings in space (in kilometers)?
gDistance(Sightings_DRWW_sp[2:4,], byid = T)

##          1        2        3
## 1 0.000000 1.316181 3.637082
## 2 1.316181 0.000000 2.386592
## 3 3.637082 2.386592 0.000000

# How far away are these sightings in time (in minutes)?
Sightings_DRWW_sp[3:4,]$WS_TIME-Sightings_DRWW_sp[2:3,]$WS_TIME

## [1] "18M 0S" "9M 0S"

# Could this be the same animal? How fast is this implied movement?
gDistance(Sightings_DRWW_sp[2:4,], byid = T)[cbind(c(1,2),c(2:3))] / 
  (c(18, 9)/60)

## [1]  4.38727 15.91061

This hypothesis would imply a traveling speed of between 4km/h and 16km/h. Is this plausible? Since fin whales are known for their speed and can sustain much higher speeds (> 30km/h), we will discard all repeated sightings each day. We will keep only the first sighting made each day by each company.

Note that there is no overlap between the sighting locations from the two whale-watch companies, indicating that an assumption of independent encounters between the two companies could be reasonable.

Sightings_DRWW_sp$PLATFORM <- as.factor(Sightings_DRWW_sp$PLATFORM)
ggplot() + gg(Domain) + gg(Sightings_DRWW_sp, colour=Sightings_DRWW_sp$PLATFORM_CODE)

Thus we subset the data to keep only the initial sightings from each company:

Sightings_Brier_nodup <- Sightings_DRWW_sp[Sightings_DRWW_sp$PLATFORM=='BRIER ISLAND WHALEWATCH',]
Sightings_Brier_nodup <- Sightings_Brier_nodup[!duplicated(Sightings_Brier_nodup$WS_DATE),]

Sightings_Quoddy_nodup <- Sightings_DRWW_sp[Sightings_DRWW_sp$PLATFORM=='QUODDY LINK',]
Sightings_Quoddy_nodup <- Sightings_Quoddy_nodup[!duplicated(Sightings_Quoddy_nodup$WS_DATE),]

dim(Sightings_DRWW_sp)[1]; dim(Sightings_Brier_nodup)[1]; dim(Sightings_Quoddy_nodup)[1];

## [1] 567

## [1] 85

## [1] 144

Notice that we have a total of 229 whale watch sightings after the subsetting procedure, down from an original count of 567 sightings.

Strictly speaking, for a log-Gaussian Cox process model to be reasonable, it is required that the initial daily encounter locations from the whale watch vessels are independent realizations from \(\lambda_{true}\). We will assume that the typical journeys/routes of the whale watch vessels remains constant across months and years under study. Thus, we assume that \(\lambda_{eff}\) is constant through time.

Is summer species density changing through time?

As we discussed earlier, we are unable to model temporal changes in the species density due to the spatially non-overlapping survey efforts across the months and years. Thus any changes in the species density from June - August, or across the years 2007-2009 will be missed. Thus, we are forced to assume that the summer species density remains constant between 2007-2009. It is the summer species density that is our target for inference.

How reasonable is our assumption that the species density remains constant across the years and across the months? Again, we can visually assess the suitability of the assumption with a plot.

If each sighting is indeed an independent realisation from a temporally static point process, then there should be no association between the spatial distances between the whale watch sightings and how far apart in time they were made.

To distances against time intervals, we first compute the distances in time and the euclidean distances in space between each sighting.

difftime_Brier<- dist(as.numeric(ymd(Sightings_Brier_nodup$WS_DATE)))
diffspace_Brier <- dist(Sightings_Brier_nodup@coords)
difftime_Brier_fac <- as.factor(difftime_Brier)

difftime_Quoddy <- dist(as.numeric(ymd(Sightings_Quoddy_nodup$WS_DATE)))
diffspace_Quoddy <- dist(Sightings_Quoddy_nodup@coords)
difftime_Quoddy_fac <- as.factor(difftime_Quoddy)

We will create multiple plots with days on the x-axis and km on the y-axis. The first plot restricted to only consider temporal differences of less than 85 days.

ggplot(data=data.frame(difftime = as.numeric(difftime_Brier)[as.numeric(difftime_Brier)<85],
                       diffspace = as.numeric(diffspace_Brier)[as.numeric(difftime_Brier)<85]),
       aes(x=difftime, y=diffspace)) +
  geom_point() + geom_smooth(method='loess') +
  xlab('difference in time (days)') +
  ylab('difference in space (km)')

This first plot indicates some moderate autocorrelation between the sightings, with encounters made ~1 month apart typically being around 50% further away than those made 24 hours apart. Notice that the trend plateaus after ~14 days. Perhaps this trend is due to the autocorrelation being caused exclusively by persistence in animal movement? Furthermore, the flat-line trend between day ~30 and day ~75 could be evidence in favour of an approximately constant density across the summer months. If the space use did indeed change dramatically across the months (e.g. due to migration), then we would expect to see a monotonically increasing trend over time.

The second plot investigate the space-time relationship across years.

ggplot(data=data.frame(difftime = as.numeric(difftime_Brier)[],
                       diffspace = as.numeric(diffspace_Brier)[]),
       aes(x=difftime, y=diffspace)) +
  geom_point() + geom_smooth(method='loess')+
  xlab('difference in time (days)') +
  ylab('difference in space (km)')

This second plot shows no evidence that the average spatial separation between whale watch sightings differs year-to-year. This supports our assumption that the spatial density can be assumed to be constant across the years, albeit within the (very) small spatial region visited by the whale watch boats departing from Brier Island.

JOE: What’s the goal of this plot?

ggplot(data=data.frame(difftime = 
difftime_Brier_fac[as.numeric(difftime_Brier)<85],
                       diffspace = as.numeric(diffspace_Brier)[as.numeric(difftime_Brier)<85]),
       aes(y=diffspace, group=difftime)) +
  geom_boxplot() +
  xlab('difference in time (days)') +
  ylab('difference in space (km)')

We now repeat the plots, but for the whale watch boats that depart from Quoddy.

ggplot(data=data.frame(difftime = as.numeric(difftime_Quoddy)[as.numeric(difftime_Quoddy)<85],
                       diffspace = as.numeric(diffspace_Quoddy)[as.numeric(difftime_Quoddy)<85]),
       aes(x=difftime, y=diffspace)) +
  geom_point() + geom_smooth(method='loess')

ggplot() + gg(Domain) + gg(Sightings_DRWW_sp, colour=Sightings_DRWW_sp$PLATFORM_CODE)

ggplot(data=data.frame(difftime = as.numeric(difftime_Quoddy)[as.numeric(difftime_Quoddy)<85 & as.numeric(diffspace_Quoddy)<50],
                       diffspace = as.numeric(diffspace_Quoddy)[as.numeric(difftime_Quoddy)<85 & as.numeric(diffspace_Quoddy)<50]),
       aes(x=difftime, y=diffspace)) +
  geom_point() + geom_smooth(method='loess')

ggplot(data=data.frame(difftime = 
difftime_Quoddy_fac[as.numeric(difftime_Quoddy)<85 & as.numeric(diffspace_Quoddy)<50],
                       diffspace = as.numeric(diffspace_Quoddy)[as.numeric(difftime_Quoddy)<85 & as.numeric(diffspace_Quoddy)<50]),
       aes(y=diffspace, group=difftime)) +
  geom_boxplot() +
  xlab('difference in time (days)') +
  ylab('difference in space (km)')

In the first plot we see a large separation in the y-axis. This is caused by a cluster of sightings made way East of port. This is seen in the second plot as the right-most cluster of blue points. To stop this from impacting our estimate of trend, we restrict the spatial differences and times to be those within each of the two clusters of sightings. The third plot displays the results from doing this. Once again, we see some potential evidence of autocorrelation upto ~14 days. This could be due to persistence of animal movement.

We have seen some evidence that residual autocorrelations remain in the sightings data. This is despite subsetting the data to the initial daily sightings. The result of this may be overdispersion. We may be able to partially correct for this by using of a log-Gaussian Cox process. By including latent effects (e.g. spatial fields) in the model, we can hopefully capture some of this additional clustering. in fact, log-Gaussian Cox processes are always overdispersed relative to Poisson processes.

So, we continue on with our exploratory analysis. We now subset the data into a training and test set. We choose the training set to contain all the sightings from 2007-2009. For the test data, we choose all the sightings from 2011.

Could we not also put 2011’s whale watch data into the training set? Below is some (optional) code showing that we cannot. A whale is sighted by at least one whale watch company on every day in 2011 that a survey detects a whale. The plot below shows that these sightings could have been of the same animal. Even if they weren’t, the tracklines in 2011 still visited areas that the whale watch vessels were present. Since our training and test datasets are required to be independent of each other, we must therefore also remove the 2011 whale watch sightings from the training data.

# what dates were survey sightings made on in 2011?
unique(Sightings_survey$DATE_LO[Sightings_survey$YEAR==2011])

## [1] "2011-06-21" "2011-08-12" "2011-08-13"

# Are sightings by the WW vessels made on these dates?
sum(grepl(Sightings_survey$DATE_LO[Sightings_survey$YEAR==2011][1],
     x=c(Sightings_Quoddy_nodup$WS_DATE[Sightings_Quoddy_nodup$YEAR==2011],
         Sightings_Brier_nodup$WS_DATE[Sightings_Brier_nodup$YEAR==2011])))>0

## [1] TRUE

sum(grepl(Sightings_survey$DATE_LO[Sightings_survey$YEAR==2011][2],
     x=c(Sightings_Quoddy_nodup$WS_DATE[Sightings_Quoddy_nodup$YEAR==2011],
         Sightings_Brier_nodup$WS_DATE[Sightings_Brier_nodup$YEAR==2011])))>0

## [1] TRUE

sum(grepl(Sightings_survey$DATE_LO[Sightings_survey$YEAR==2011][3],
     x=c(Sightings_Quoddy_nodup$WS_DATE[Sightings_Quoddy_nodup$YEAR==2011],
         Sightings_Brier_nodup$WS_DATE[Sightings_Brier_nodup$YEAR==2011])))>0

## [1] TRUE

# Could they be of the same animal? 
ggplot() + gg(Domain) + 
  gg(Sightings_survey[Sightings_survey$YEAR==2011,],colour='green') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR==2011,],colour='yellow')

Below we split the data into training and test data.

Sightings_Brier_nodup_test <- Sightings_Brier_nodup[Sightings_Brier_nodup$YEAR==2011,]
Sightings_Brier_nodup <- Sightings_Brier_nodup[Sightings_Brier_nodup$YEAR!=2011,]
Sightings_Quoddy_nodup_test <- Sightings_Quoddy_nodup[Sightings_Quoddy_nodup$YEAR==2011,]
Sightings_Quoddy_nodup <- Sightings_Quoddy_nodup[Sightings_Quoddy_nodup$YEAR!=2011,]
Sightings_survey_test <- Sightings_survey[Sightings_survey$YEAR==2011,]
Sightings_survey <- Sightings_survey[Sightings_survey$YEAR!=2011,]
Effort_survey_test <- Effort_survey[Effort_survey$YEAR=='2011',]
Effort_survey <- Effort_survey[Effort_survey$YEAR!='2011',]

# How many survey sightings by year?
table(Sightings_survey$YEAR)

## 
## 2007 2008 2009 
##   45   17    1

Next, we evaluate a crude estimate of the total amount of survey effort spent each year, as measured by the total trackline length. Using this, we then crudely estimate a ‘catch per unit effort’ measure by year.

# How much survey effort (in trackline length) is there by year
by(gLength(Effort_survey,byid = T), Effort_survey$YEAR, sum)

## Effort_survey$YEAR: 2007
## [1] 11643.15
## ------------------------------------------------------------ 
## Effort_survey$YEAR: 2008
## [1] 571.9083
## ------------------------------------------------------------ 
## Effort_survey$YEAR: 2009
## [1] 819.9337

# Crude estimate of relative `CPUE'
(table(Sightings_survey$YEAR) / 
       by(gLength(Effort_survey,byid = T), Effort_survey$YEAR, sum))/
  min(table(Sightings_survey$YEAR) / 
        by(gLength(Effort_survey,byid = T), Effort_survey$YEAR, sum))

## 
##      2007      2008      2009 
##  3.168989 24.372565  1.000000

# How many WW sightings by year
table(Sightings_Quoddy_nodup$YEAR)

## 
## 2007 2008 2009 
##   40   24   44

table(Sightings_Brier_nodup$YEAR)

## 
## 2007 2008 2009 
##   24   19   19

The ‘CPUE’ values are highly variable! The wild differences could be due to the small efforts in years 2008 and 2009 relative to 2007. Alternatively, the differences could actually reflect real differences in whale density across the three distinct regions visited each year! Without taking a model-based approach and controlling for differences in spatial effort, it is unclear how to combine the sightings from these three surveys.

By taking a model-based approach to inference, we are able to combine the data from all three surveys to estimate the whale density across space.

ggplot() +
  gg(Domain) +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2007',], colour='red') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2008',], colour='black') +
  gg.spatiallines_mod(Effort_survey[Effort_survey$YEAR=='2009',], colour='blue') +
  gg.spatiallines_mod(Effort_survey_test, colour='purple')

Note that the survey effort in 2011 (in purple) overlaps with all the previous years’ surveys and WW data. 2011 therefore provides a reasonable test dataset for comparing models. The 3 years of surveys were conducted under 3 different surveys following similar (but different) protocols. Unfortunately, due to the lack of spatial overlap between the surveys’ efforts (tracklines), we are unable to account for any differences between the survey protocols within a model without strong prior information available. For example, an intercept added to the model for each survey would be confounded by the spatial field.

xtabs(~DATASET+YEAR,Effort_survey@data)

##         YEAR
## DATASET  2007 2008 2009
##   DFO      49    0    0
##   NOAA_1    3   20   19
##   NOAA_2   49    0    0

Graphically investigating trends between the environmental covariates and the whale density

We now investigate trends between the whale density and the covariates bathymetry and Slope. A simple first step is to evaluate the empirical densities of each covariate at both the encounter locations and across the domain. One easy and quick way to do this is to convert the SpatialPixelsDataFrame objects into objects of class im from the spatstat package. Then, after creating a ppp point pattern object containing the sightings locations from all sources, we can easily extract the values of each covariate at each point location. Alternatively, we can use the function over() from the sp package. We choose to create im objects so that we can later use the ppm function from spatstat for exploratory analysis purposes. ppm fits Poisson process models, equivalent to Maxent models.

# Convert covariates to im class
Slope_im <- as(as(Slope, 'SpatialGridDataFrame'),'im')
Bathym_im <- as(as(Bathym, 'SpatialGridDataFrame'),'im')
# Convert Observation locations to ppp object
All_obs_ppp <- ppp(x=c(Sightings_survey@coords[,1],
                       Sightings_Brier_nodup@coords[,1],
                       Sightings_Quoddy_nodup@coords[,1]),
                   y=c(Sightings_survey@coords[,2],
                       Sightings_Brier_nodup@coords[,2],
                       Sightings_Quoddy_nodup@coords[,2]),
                   marks = as.factor(c(rep('survey',dim(Sightings_survey@coords)[1]),
            rep('Brier',dim(Sightings_Brier_nodup@coords)[1]),
            rep('Quoddy',dim(Sightings_Quoddy_nodup@coords)[1]))),
                   window = as(Domain, 'owin'))
# Notice how easy it is to extract the values of slope at these locations!
Slope_obs <- Slope_im[All_obs_ppp]
ggplot(data=data.frame(Slope_obs = Slope_obs),
       aes(x=Slope_obs)) +
  geom_density() +
  geom_density(data=data.frame(Slope_dom = as.numeric(Slope_im$v)),
            aes(x=Slope_dom), colour='red') + xlab('Slope')

# Perhaps a log transform is desirable here to make the covariate more 'normal'
ggplot(data=data.frame(Slope_obs = log(Slope_obs)),
       aes(x=Slope_obs)) +
  geom_density() +
  geom_density(data=data.frame(Slope_dom = log(as.numeric(Slope_im$v))),
               aes(x=Slope_dom), colour='red') + xlab('log Slope')

# Perhaps there is evidence of whales preferring areas of higher slope?

Bathym_obs <- Bathym_im[All_obs_ppp]
ggplot(data=data.frame(Bathym_obs = Bathym_obs),
       aes(x=Bathym_obs)) +
  geom_density() +
  geom_density(data=data.frame(Bathym_dom = as.numeric(Bathym_im$v[Bathym_im$v<0])),
               aes(x=Bathym_dom), colour='red') + xlab('Bathymetry (m)')

# Perhaps a log transform is desired here. Max bathym is 313
# Lets only consider values in water
# Flip the sign to make it depth (larger is deeper)
ggplot(data=data.frame(Bathym_obs = log(1-Bathym_obs)),
       aes(x=Bathym_obs)) +
  geom_density() +
  geom_density(data=data.frame(Bathym_dom = log(as.numeric(1-Bathym_im$v[Bathym_im$v<0]))),
               aes(x=Bathym_dom), colour='red') +
  geom_vline(xintercept = 0, colour='blue') + xlab('log Depth')

# Perhaps there is evidence of whales preferring areas of shallower waters?

For each covariate, we plot the empirical density at the sightings locations in black and at the background in red. From this, we see that sightings locations are associated with regions with shallower waters and flatter slopes compared with the domain as a whole. However, these results may simply reflect the types of regions preferentially visited by observers!

Next, we could go one step further and fit Poisson point process models to the sightings data. These are equivalent to Maxent models. Ignoring effort for the time being, what does a Maxent model tell us?

log_Slope_im <- Slope_im
log_Slope_im$v <- log(log_Slope_im$v)
log_Bathym_im <- Bathym_im
log_Bathym_im[log_Bathym_im >= 0] <- -1
log_Bathym_im$v <- log(1-log_Bathym_im$v)
Maxent_mod <- ppm(All_obs_ppp~log_Slope_im++log_Bathym_im,
                  data=list(log_Slope_im=log_Slope_im, log_Bathym_im=log_Bathym_im,
                            Domain=as(Domain, 'owin')),
                  subset = Domain)
summary(Maxent_mod)

## Point process model
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = All_obs_ppp ~ log_Slope_im + +log_Bathym_im, 
##     data = list(log_Slope_im = log_Slope_im, log_Bathym_im = log_Bathym_im, 
##         Domain = as(Domain, "owin")), subset = Domain)
## Edge correction: "border"
##  [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
## 
## Data pattern:
## Marked planar point pattern:  232 points
## Average intensity 0.000633 points per square unit
## Multitype:
##        frequency proportion intensity
## Brier         62      0.267  0.000169
## Quoddy       107      0.461  0.000292
## survey        63      0.272  0.000172
## 
## Window: polygonal boundary
## 2 separate polygons (1 hole)
##                  vertices      area relative.area
## polygon 1            1519 366946.00      1.000000
## polygon 2 (hole)       38   -179.42     -0.000489
## enclosing rectangle: [112.6639, 1141.4561] x [4451.802, 5259.268] units
##                      (1029 x 807.5 units)
## Window area = 366766 square units
## Fraction of frame area: 0.442
## 
## Dummy quadrature points:
##      64 x 64 grid of dummy points, plus 4 corner points
##      dummy spacing: 16.07488 x 12.61666 units
## 
## Original dummy parameters: =
## Marked planar point pattern:  6191 points
## Average intensity 0.0169 points per square unit
## Multitype:
##        frequency proportion intensity
## Brier       2080      0.336   0.00567
## Quoddy      2030      0.329   0.00555
## survey      2080      0.336   0.00567
## 
## Window: polygonal boundary
## 2 separate polygons (1 hole)
##                  vertices      area relative.area
## polygon 1            1519 366946.00      1.000000
## polygon 2 (hole)       38   -179.42     -0.000489
## enclosing rectangle: [112.6639, 1141.4561] x [4451.802, 5259.268] units
##                      (1029 x 807.5 units)
## Window area = 366766 square units
## Fraction of frame area: 0.442
## Quadrature weights:
##      (counting weights based on 64 x 64 array of rectangular tiles)
## All weights:
##  range: [2.6, 203]   total: 1100000
## Weights on data points:
##  range: [2.6, 101]   total: 6790
## Weights on dummy points:
##  range: [2.6, 203]   total: 1090000
## --------------------------------------------------------------------------------
## FITTED MODEL:
## 
## Nonstationary multitype Poisson process
## Possible marks:
## Brier Quoddy survey
## ---- Intensity: ----
## 
## Log intensity: ~log_Slope_im + +log_Bathym_im
## Model depends on external covariates 'log_Slope_im' and 'log_Bathym_im'
## Covariates provided:
##  log_Slope_im: im
##  log_Bathym_im: im
##  Domain: owin
## 
## Fitted trend coefficients:
##   (Intercept)  log_Slope_im log_Bathym_im 
##    -4.2918939     0.6780343    -0.7629490 
## 
##                 Estimate       S.E.    CI95.lo    CI95.hi Ztest      Zval
## (Intercept)   -4.2918939 0.17472036 -4.6343395 -3.9494482   *** -24.56436
## log_Slope_im   0.6780343 0.05974179  0.5609425  0.7951260   ***  11.34941
## log_Bathym_im -0.7629490 0.03610715 -0.8337177 -0.6921802   *** -21.13013
## 
## ----------- gory details -----
## 
## Fitted regular parameters (theta):
##   (Intercept)  log_Slope_im log_Bathym_im 
##    -4.2918939     0.6780343    -0.7629490 
## 
## Fitted exp(theta):
##   (Intercept)  log_Slope_im log_Bathym_im 
##    0.01367899    1.97000142    0.46628933

We find some evidence that the whales prefer regions of higher slope and shallower waters.

Next, we can attempt to make a crude adjustment for effort and see if these estimates change. We have already created a SpatialPixelsDataFrame object containing distance from the two whale watch ports. Next, we can create a SpatialPixelsDataFrame containing the perpendicular distances to the nearest trackline. This fails to account for overlap in survey tracklines, but offers a reasonable approach to crudely control for effort.

Dist_Surveylines <- as(Dist_Brier,'SpatialPoints')
Dist_Surveylines$Dist_surveylines <- as.numeric(apply(gDistance(Dist_Surveylines, Effort_survey,byid = T),2,min))
# Define a 'pretty' colour palette
colsc <- function(...) {
  scale_fill_gradientn(colours = rev(RColorBrewer::brewer.pal(11,"RdYlBu")),
                       limits = range(...))
}
Dist_Surveylines <- as(Dist_Surveylines,'SpatialPixelsDataFrame')
ggplot() + gg(Domain) + gg(Dist_Surveylines) + colsc(Dist_Surveylines@data[,1])

# What is the maximum detection distance?
ggplot(Sightings_survey@data, aes(x=DISTANCE)) + geom_histogram()

# Let's add a half-norm detection probability function, capped at 5km
Dist_Surveylines$Dist_surveylines[Dist_Surveylines$Dist_surveylines > 5000] <- 1e3
# need to square the value
Dist_Surveylines$Dist_surveylines <- scale(Dist_Surveylines$Dist_surveylines^2)
Dist_Brier_sq <- as(Dist_Brier,'SpatialPixels')
Dist_Brier_sq$Dist_Brier_sq <- scale(Dist_Brier$Dist_Brier^2)
Dist_Quoddy_sq <- as(Dist_Quoddy,'SpatialPixels')
Dist_Quoddy_sq$Dist_Quoddy_sq <- scale(Dist_Quoddy$Dist_Quoddy^2)

Maxent_mod <- ppm(All_obs_ppp~log_Slope_im+log_Bathym_im +
                    I(ifelse(marks=='Brier',1,0)):Dist_Brier_im +
                    I(ifelse(marks=='Quoddy',1,0)):Dist_Quoddy_im +
                    I(ifelse(marks=='survey',1,0)):Dist_Survey_im +
                    marks,
                  data=list(log_Slope_im=log_Slope_im, 
                            log_Bathym_im=log_Bathym_im,
                            Dist_Brier_im = as(as(Dist_Brier_sq,'SpatialGridDataFrame'),'im'),
                            Dist_Quoddy_im = as(as(Dist_Quoddy_sq,'SpatialGridDataFrame'),'im'),
                            Dist_Survey_im = as(as(Dist_Surveylines,'SpatialGridDataFrame'),'im'),
                            Domain=as(Domain, 'owin')),
                  subset = Domain)
coef(summary(Maxent_mod))

##                                                       Estimate         S.E.
## (Intercept)                                       -820.4577108 109.50202223
## log_Slope_im                                         0.4054561   0.06224574
## log_Bathym_im                                        0.1593891   0.07068249
## marksQuoddy                                        482.3514087 113.06812329
## markssurvey                                        547.9081559 119.51263464
## I(ifelse(marks == "Brier", 1, 0)):Dist_Brier_im   -687.6788840  92.23298929
## I(ifelse(marks == "Quoddy", 1, 0)):Dist_Quoddy_im -245.1096937  20.37752448
## I(ifelse(marks == "survey", 1, 0)):Dist_Survey_im -643.2218902 117.20897718
##                                                         CI95.lo      CI95.hi
## (Intercept)                                       -1.035078e+03 -605.8376910
## log_Slope_im                                       2.834567e-01    0.5274555
## log_Bathym_im                                      2.085397e-02    0.2979242
## marksQuoddy                                        2.607420e+02  703.9608582
## markssurvey                                        3.136677e+02  782.1486155
## I(ifelse(marks == "Brier", 1, 0)):Dist_Brier_im   -8.684522e+02 -506.9055468
## I(ifelse(marks == "Quoddy", 1, 0)):Dist_Quoddy_im -2.850489e+02 -205.1704797
## I(ifelse(marks == "survey", 1, 0)):Dist_Survey_im -8.729473e+02 -413.4965163
##                                                   Ztest       Zval
## (Intercept)                                         ***  -7.492626
## log_Slope_im                                        ***   6.513797
## log_Bathym_im                                         *   2.255001
## marksQuoddy                                         ***   4.266025
## markssurvey                                         ***   4.584521
## I(ifelse(marks == "Brier", 1, 0)):Dist_Brier_im     ***  -7.455888
## I(ifelse(marks == "Quoddy", 1, 0)):Dist_Quoddy_im   *** -12.028433
## I(ifelse(marks == "survey", 1, 0)):Dist_Survey_im   ***  -5.487821

With this rough attempt at controlling for heterogeneous effort, we see that both log slope and log depth are found to be significantly associated with the whale density, with positive coefficients for both. The negative effects of all three complicated expressions imply that a negative association between ‘distance’ and observed intensity is witnessed (agreeing with common sense).

We can plot the true whale intensity predicted from this model:

# remove the effects of observers for plotting
pred_coef <- coef(Maxent_mod)
pred_coef[c(1,4,5,6,7,8)] <- 0
plot(predict.ppm(Maxent_mod, ngrid=c(300,300),
             new.coef=pred_coef)[[1]],superimpose=F,
     hcl.colors(60, "YlOrRd", rev = TRUE))

Alternatively, we can also plot the fitted observed intensity surface \(\lambda_{obs}\) (=\(\lambda_{true} \times \lambda_{effort}\)) for all observer types. We plot the observed intensity surfaces for both the survey and Quoddy-based whale watch vessels.

pred_coef <- coef(Maxent_mod)
plot(log(predict(Maxent_mod, ngrid=c(300,300),
             new.coef=pred_coef)[[3]]),superimpose=F,
     hcl.colors(60, "YlOrRd", rev = TRUE))

# Plot Quoddy
plot(log(predict(Maxent_mod, ngrid=c(300,300),
             new.coef=pred_coef)[[2]]),superimpose=F,
     hcl.colors(60, "YlOrRd", rev = TRUE))

This exploratory analysis is useful at informing us about possible species-environment relationships. By playing around with different covariates and functional forms (e.g. quadratic effects), we can develop an understanding of which models to formally compare in the modelling stage.

Unfortunately, by being in the inhomogeneous Poisson process class of point processes, the above models lack one crucial feature. They are unable to account for any additional spatial correlations/clustering that are frequently caused by unmeasured environmental covariates and biological processes acting behind the scenes.

We can actually investigate the suitability (or unsuitability) of this inhomogeneous Poisson process assumption by plotting the (estimated) inhomogeneous K functions. Loosely speaking, for a homogeneous Poisson process (with constant intensity) the K function evaluated at a distance r defines a (scalar multiple of) the expected number of extra events that occur within a distance \(r\) of a randomly chosen point (event). The K function can be evaluated exactly for homogeneous Poisson processes and takes the form \(K(r) = \pi r^2\). The empirical K function can also be computed from the data and compared with the true one. The empirical K function counts the observed number of inter-point distances (in the observed data) that fall within a sequence of distances \(r_1, r_2 ,..., r_m\). Plotting the observed (empirical) and expected (true) K functions together allows for model assumptions to be assessed.

If the empirical K function falls ‘significantly’ above the expected curve at a distance \(r\), then additional clustering has been detected in the data that cannot be explained by the homogeneous model alone. Conversely, if the empirical K function falls ‘significantly’ below the expected curve at a distance \(r\), then additional repulsion or regularity has been detected in the data that cannot be explained by the homogeneous Poisson model alone. This clustering or repulsion may be caused by covariates or biological processes that have been missed!

The K function and empirical K function can also be computed for inhomogeneous Poisson process (or MAXENT) models. The interpretation of the plots is similar, with large discrepancies providing evidence of additional unexplained clustering/repulsion existing in the data that may be caused by unmeasured covariates. Let’s plot these two curves now!

plot(Kinhom(All_obs_ppp,lambda=Maxent_mod,correction = 'best',normpower = 1))

We see a large discrepancy between the true curve (in red) and the empirical curve in black. Thus, the fitted model in its current form is inadequate.

This is why we turn to log-Gaussian Cox processes in the next workshop. By assuming the intensity surface to be a realisation from a spatially-smooth (log-Gaussian) random field, we are able to adjust our inference to take into account any additional spatial correlations and/or unmeasured covariates! By doing this, both parameter estimates and measures of uncertainty should be better adjusted to account for these autocorrelations.

Exercises

If you got stuck on any of the exercises, then please feel free to try them again. Here are links to the problems:

1. Plotting data by year
2. Defining a custom colour palette in ggplot for spatial plots

Bonus Exercises

1. Transform one of the spatial objects back into longitude latitude.
2. Transform one of the spatial objects into units of meters.
3. (More challenging) Investigate the relationship of distance from port with the two sources of whale watch data? What do you notice? What functional form could explain this type of behaviour? (Hint: use the over() function to extract the values of a spatial covariate (of type SpatialPixelsDataFrame) at point locations (stored as a SpatialPointsDataFrame object). Remember to subset the data accordingly!)

Additional tips and code

Using OpenStreetMaps instead of Google Maps

For publication, there can be issues regarding copyright of Google Maps. Using OpenStreetMap can help. To guarantee this simply add the following argument: source='osm', force=TRUE to gmap(). Double check the console that the maps are indeed being downloaded from stamen or osm. For brevity we have suppressed the messages.

gmap(Sightings_survey, source='osm', force=TRUE) +
  gg(Domain) +
  gg.spatiallines_mod(Effort_survey) +
  gg(Sightings_survey, colour='blue') +
  gg(Sightings_DRWW_sp, colour='purple') +
  gg(WW_ports, colour='red')

Note for this to work it needs to be in the original project (lat/lon), not UTM.

The multiplot() function is a very flexible function that enables publication-quality figures to be made with relative ease.

Changing the layout of multiplot()

To change the order of the plots you can change the argument byrow=TRUE to byrow=FALSE.

multiplot(ggplot() +
  gg(Domain) +
  gg(Bathym) + xlab('East(km)') + ylab('North(km)') + labs(fill='Bathymetry'),
ggplot() +
  gg(Domain) +
  gg(Slope) + xlab('East(km)') + ylab('North(km)') + labs(fill='Slope'),
ggplot() +
  gg(Domain) +
  gg(Dist_Brier) + xlab('East(km)') + ylab('North(km)'),
ggplot() +
  gg(Domain) +
  gg(Dist_Quoddy) + xlab('East(km)') + ylab('North(km)'),
layout=matrix(1:4, nrow=2, ncol=2, byrow = FALSE))

We can also change the size of the figures by changing the matrix in the layout.

multiplot(ggplot() +
  gg(Domain) +
  gg(Bathym) + xlab('East(km)') + ylab('North(km)') + labs(fill='Bathymetry'),
ggplot() +
  gg(Domain) +
  gg(Slope) + xlab('East(km)') + ylab('North(km)') + labs(fill='Slope'),
ggplot() +
  gg(Domain) +
  gg(Dist_Brier) + xlab('East(km)') + ylab('North(km)'),
ggplot() +
  gg(Domain) +
  gg(Dist_Quoddy) + xlab('East(km)') + ylab('North(km)'),
layout=matrix(c(1,1,2,3,4,4), nrow=3, ncol=2, byrow = TRUE))

As you can see plots of different size stretches the maps, to keep the set projection we can use coord_fixed(ratio = 1).

multiplot(ggplot() +
  gg(Domain) +
  gg(Bathym) + xlab('East(km)') + ylab('North(km)') + labs(fill='Bathymetry') +
    coord_fixed(ratio = 1),
ggplot() +
  gg(Domain) +
  gg(Slope) + xlab('East(km)') + ylab('North(km)') + labs(fill='Slope') +
  coord_fixed(ratio = 1),
ggplot() +
  gg(Domain) +
  gg(Dist_Brier) + xlab('East(km)') + ylab('North(km)') + 
  coord_fixed(ratio = 1),
ggplot() +
  gg(Domain) +
  gg(Dist_Quoddy) + xlab('East(km)') + ylab('North(km)') + 
  coord_fixed(ratio = 1),
layout=matrix(c(1,1,2,3,4,4), nrow=3, ncol=2, byrow = TRUE))

Acknowledgements

The code for hiding the code chunks came from Martin Schmelzer, found here